Check if you maybe have a stuck Couatl64_msfs.exe running in the Taskbar. It should have been fixed by restarting windows, of course, but it's something to check.
After what you said, I checked to see if 'Couatl64_msfs.exe' and it wasn't working in the taskbar. I found that when I was running MSFS and appeared in an airport, it also did not work and did not appear in the taskbar.
What did I do about it? I turned off MSFS and ran Couatl first. It was still giving me the 'connecting to simconnect' message.
I opened MSFS and launched my flight at an airport. Since I was running Couatl before, the Couatl in the taskbar was also visible, and when I selected restart from here, an extended log file was filled in, and the GSX menus were opened. I tried at four different airports without ever closing msfs, and I saw that the GSX menus open and work without any problems.
After all this, I turned off MSFS, restarted my computer, opened MSFS without running Couatl, and started my plane at an airport with the same problem again. The Couatl taskbar is still not visible in the taskbar, and I keep getting the 'Couatl engine hasn't started' error. Couatl.exe does not start by itself, and I have to intervene manually every time to use GSX.
I attached a new log file.